A drone's GPS doesn't say "turn slightly left": it gives two numbers, a position $(x, y)$. And when the drone flies from one point to another, that flight can also be described with two numbers: how far it moves horizontally and how far it moves vertically. We call that "flight" a vector, and with it you will solve almost everything in this topic: distances, midpoints, lines…
You already know the Cartesian plane from earlier years: two perpendicular axes, and a point $P(x, y)$ is plotted by counting $x$ horizontally and $y$ vertically from the origin $O(0,0)$. What is new in this topic is that you will use this map for more than plotting points: you will measure and calculate on it.
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Find the exact distance between the points $P(1, 4)$ and $Q(-5, -4)$ in the Cartesian plane.
Find the exact distance between the points $P(1, 4)$ and $Q(-5, -4)$ in the Cartesian plane.
The vector $\vec{AB} = (v_1, v_2)$ forms a right-angled triangle with the axes: $v_1$ and $v_2$ are the legs. By Pythagoras, its magnitude (length) is:
$$|\vec v| = \sqrt{v_1^2 + v_2^2}$$
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Determine the magnitude of the vector $\vec{v} = (6, -8)$. Simplify the result if possible.
Determine the magnitude of the vector $\vec{v} = (6, -8)$. Simplify the result if possible.
A forest surveillance drone takes off from point $A(-2, 1)$ on a grid map with units in kilometres. It flies in a straight line to point $B(4, 9)$ to inspect a…
A forest surveillance drone takes off from point $A(-2, 1)$ on a grid map with units in kilometres. It flies in a straight line to point $B(4, 9)$ to inspect a suspicious clearing, and from there moves directly to point $C(10, 1)$ to examine a shelter. Calculate the total distance covered by the drone during this journey.
The geometric centre of a circular ice rink is located at coordinates $C(2, -3)$. If one end of the rink, where the entrance gate is located, is at point $A(-1, 1)$,…
The geometric centre of a circular ice rink is located at coordinates $C(2, -3)$. If one end of the rink, where the entrance gate is located, is at point $A(-1, 1)$, find the coordinates of the diametrically opposite end $B$ and calculate the total diameter of the rink.
An architect is designing a suspension bridge in which three of the main cable anchors must be perfectly collinear to ensure correct tension. On the construction plan,…
An architect is designing a suspension bridge in which three of the main cable anchors must be perfectly collinear to ensure correct tension. On the construction plan, their coordinates are $P(-3, 2)$, $Q(1, 4)$ and $R(a, 7)$. Calculate the value of parameter $a$ so that the structure is stable.